I do think the cable will need to be either constantly growing at one end and eaten at the other, or little robots might need to go up and down constantly eating and remaking it, or possibly both.

The cable will also need to have tweaked characteristics along the whole length because of different enviroments. The lenght of exposure to sunlight compared to shade, radiation levels, being exposed to different parts of the atmosphere etc.

Getting a strong enough fibre is probably not enough to make the cable, it is barely the start.